A string vibrates with a frequency of 200 ~Hz . When its length is doubled and tension is altered, it begins to vibrate with a frequency of 300 ~Hz . The ratio of the new tension to the original tension is

Step-by-step solution

The fundamental frequency of a vibrating string is given by f = 1 2L T , where L is the length, T is the tension, and is the linear mass density. Let the initial frequency be f₁ = 200 Hz , length be L₁ = L , and tension be T₁ . Then f₁ = 1 2L T₁ . Let the new frequency be f₂ = 300 Hz , new length be L₂ = 2L , and new tension be T₂ . Then f₂ = 1 2(2L) T₂ = 1 4L T₂ . Taking the ratio of the two frequencies: f₂ f₁ = 1 4L T₂ 1 2L T₁ = 1 2 T₂ T₁ . Substituting the given values: 300 200 = 1 2 T₂ T₁ 3 2 = 1 2 T₂ T₁ . Mult